on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
A monoidal model category is a model category which is also a closed monoidal category in a compatible way. In particular, its homotopy category inherits a closed monoidal structure.
A (symmetric) monoidal model category is model category $\mathcal{C}$ equipped with the structure of a closed symmetric monoidal category $(\mathcal{C}, \otimes, I)$ such that the following two compatibility conditions are satisfied
(pushout-product axiom) For every pair of cofibrations $f \colon X \to Y$ and $f' \colon X' \to Y'$, their pushout-product, hence the induced morphism out of the cofibered coproduct over ways of forming the tensor product of these objects
is itself a cofibration, which, furthermore, is acyclic if $f$ or $f'$ is.
(Equivalently this says that the tensor product $\otimes : C \times C \to C$ is a left Quillen bifunctor.)
(unit axiom) For every cofibrant object $X$ and every cofibrant resolution $\emptyset \hookrightarrow Q I \stackrel{p_I}{\longrightarrow} \ast$ of the tensor unit $I$, the resulting morphism
is a weak equivalence.
The pushout-product axiom in def. 1 implies that for $X$ a cofibrant object, then the functor $X \otimes (-)$ is preserves cofibrations and acyclic cofibrations.
In particular if the tensor unit $I$ happens to be cofibrant, then the unit axiom in def. 1 is implied by the pushout-product axiom. (Because then $Q I \to I$ is a weak equivalence between cofibrant objects and such are preserved by functors that preserve acyclic cofibrations, by Ken Brown's lemma. )
We say a monoidal model category, def. 1, satisfies the monoid axiom, def. 1, if every morphism that is obtained as a transfinite composition of pushouts of tensor products $X\otimes f$ of acyclic cofibrations $f$ with any object $X$ is a weak equivalence.
(Schwede-Shipley 00, def. 3.3.).
In particular, the axiom in def. 2 says that for every object $X$ the functor $X \otimes (-)$ sends acyclic cofibrations to weak equivalences.
Let $(\mathcal{C}, \otimes, I)$ be a monoidal model category. Then the left derived functor of the tensor product exists and makes the homotopy category into a monoidal category $(Ho(\mathcal{C}), \otimes^L, \gamma(I))$.
If in in addition $(\mathcal{C}, \otimes)$ satisfies the monoid axiom, then the localization functor $\gamma\colon \mathcal{C}\to Ho(\mathcal{C})$ carries the structure of a lax monoidal functor
Consider the explicit model of $Ho(\mathcal{C})$ as the category of fibrant-cofibrant objects in $\mathcal{C}$ with left/right-homotopy classes of morphisms between them (discussed at homotopy category of a model category).
A derived functor exists if its restriction to this subcategory preserves weak equivalences. Now the pushout-product axiom implies that on the subcategory of cofibrant objects the functor $\otimes$ preserves acyclic cofibrations, and then the preservation of all weak equivalences follows by Ken Brown's lemma.
Hence $\otimes^L$ exists and its associativity follows simply by restriction. It remains to see its unitality.
To that end, consider the construction of the localization functor $\gamma$ via a fixed but arbitrary choice of (co-)fibrant replacements $Q$ and $R$, assumed to be the identity on (co-)fibrant objects. We fix notation as follows:
Now to see that $\gamma(I)$ is the tensor unit for $\otimes^L$, notice that in the zig-zag
all morphisms are weak equivalences: For the first two this is due to the pushout-product axiom, for the third this is due to the unit axiom on a monoidal model category. It follows that under $\gamma(-)$ this zig-zig gives an isomorphism
and similarly for tensoring with $\gamma(I)$ from the right.
To exhibit lax monoidal structure on $\gamma$, we need to construct a natural transformation
and show that it satisfies the the appropriate associativity and unitality condition.
By the definitions at homotopy category of a model category, the morphism in question is to be of the form
To this end, consider the zig-zag
and observe that the two morphisms on the left are weak equivalences, as indicated, by the pushout-product axiom satisfied by $\otimes$.
Hence applying $\gamma$ to this zig-zag, which is given by the two horizontal part of the following digram
and inverting the first two morphisms, this yields a natural transformation as required.
To see that this satisfies associativity if the monoid axiom holds, tensor the entire diagram above on the right with $(R Q Z)$ and consider the following pasting composite:
Observe that under $\gamma$ the total top zig-zag in this diagram gives
Now by the monoid axiom (but not by the pushout-product axiom!), the horizontal maps in the square in the bottom right (labeled $\star$) are weak equivalences. This implies that the total horizontal part of the diagram is a zig-zag in the first place, and that under $\gamma$ the total top zig-zag is equal to the image of that total bottom zig-zag. But by functoriality of $\otimes$, that image of the bottom zig-zag is
The same argument applies to left tensoring with $R Q Z$ instead of right tensoring, and so in both cases we reduce to the same morphism in the homotopy category, thus showing the associativity condition on the transformation that exhibits $\gamma$ as a lax monoidal functor.
A nice category of topological spaces with cartesian product and the classical model structure on topological spaces.
The category of simplicial sets with cartesian product and the classical model structure on simplicial sets.
The classical model structure on pointed topological spaces or pointed simplicial sets with the smash product of pointed objects.
With respect to a symmetric monoidal smash product of spectra:
(Schwede-Shipley 00, MMSS 00, theorem 12.1 (iii) with prop. 12.3)
The standard example of a monoidal model category whose unit is not cofibrant is the category of EKMM S-modules.
The category Cat with cartesian product and the folk model structure.
The category Gray of strict 2-categories with the Gray tensor product and the Lack model structure?.
Let $\mathcal{E}$ be a category equipped with the structure of
such that
the model structure is cofibrantly generated;
the tensor unit $I$ is cofibrant.
Under these conditions there is for each finite group $G$ the structure of a monoidal model category on the category $\mathcal{E}^{\mathbf{B}G}$ of objects in $\mathcal{E}$ equipped with a $G$-action, for which the forgetful functor
preserves and reflects fibrations and weak equivalences.
See for instance (BergerMoerdijk 2.5).
See model structure on monoids in a monoidal model category.
A general standard reference is
The monoidal structures for a symmetric monoidal smash product of spectra are due to
Stefan Schwede, Brooke Shipley, Algebras and modules in monoidal model categories Proc. London Math. Soc. (2000) 80(2): 491-511 (pdf)
Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley, part III of Model categories of diagram spectra, Proceedings London Mathematical Society Volume 82, Issue 2, 2000 (pdf, publisher)
The monoidal model structure on $\mathcal{E}^{\mathbf{B}G}$ is discussed for instance in
Relation to symmetric monoidal (infinity,1)-categories is discussed in